How did they know that he was correctly identifying them?
<rimshot />
In all seriousness, though: Once they'd caught their suspect and thrown them in Newgate Prison, the authorities could always summon people from the suspect's hometown to ID them more precisely. In theory. In practice, that would surely have been very expensive and slow, and I have no idea how diligent they really were about all that pesky "legal procedure" stuff, particularly if the prisoner didn't have the money to hire a lawyer. Back then they charged prisoners for everything: you had to pay room and board in prison, so if you hadn't managed to hang onto your money (which was probably pretty difficult when you were being arrested by freelance thief-takers...) and you didn't have local friends to look you up in prison and help pay your bills, you could be stuck in there for life, like Charley on the MTA.
How do yo correctly identify someone by sight?
Everyone who can talk (and I would guess grunt or otherwise shake the vocal cords) has a voice signature. The difference in some might be minute, even undetectable by most people, but the difference is still there.
People who have lost their sight tend to pick up more on the other senses (unless there's some damage there too). Consider braille. Can you distinguish the different dots?